>>It makes sense that if the finest topology is called >>"discrete" that the coarsest could be called "indiscrete".>>I think that a trivial object would be one that is embedded >in all objects of the same type. For example, the trivial >group is the group with one element.

I think that in this case, the intent of the terminology is that,for a given set X, the trivial topology on X is the simplest possible topology on X.

However it's also true that any topology on X is a supersetof the trivial topology on X.

>If there is such a thing as "the trivial topology [without >mentioning the underlying set]" then that might be the topology >on the empty set where the only open set is the empty set.

No -- the terminology "the trivial topology" always refers toa topology on a given set X.

>If the underlying set is X, then I would think "The trivial >topology on X" is a fine way of describing the indiscrete >topology, since the open sets in that topology on X are exactly >the sets that are open in every topology on X. So it's >analogous to the "trivial group".

Yes, exactly.

>Is "trivial topology on X" a standard way of referring to the>indiscrete topology on X? If not, I think it should be.

The terminologies

"the trivial topology on X"

"the indiscrete topology on X"

are both accepted, but I think "the indiscrete topology" is the one more commonly used.